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Quantization of solitons

H. A. Haus, Yinchieh Lai

Published May 21, 1990• 0 citations
OpenAlex

Abstract

Solitons on fibers are ideal for all optical switching operations.1 Squeezed solitons promise to provide improved measurement capabilities.2 The self-induced transparency (SIT) in CdS has been proposed as a particularly convenient means of squeezing.3 For this reason it is important to understand the quantization problem of solitons. Several authors have attacked the problem in the case of the nonlinear Schrodinger equation (NSE) both numerically4 and analytically.5 A particularly simple approach to quantization can be worked out in the limit of a linearized approximation. Since all switching and squeezing operations employ pulses with many photons (n > 10), the linearization is always legitimate.

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Solitons on fibers are ideal for all optical switching operations.1 Squeezed solitons promise to provide improved measurement capabilities.2 The self-induced transparency (SIT) in CdS has been proposed as a particularly convenient means of squeezing.3 For this reason it is important to understand the quantization problem of solitons. Several authors have attacked the problem in the case of the nonlinear Schrodinger equation (NSE) both numerically4 and analytically.5 A particularly simple approach to quantization can be worked out in the limit of a linearized approximation. Since all switching and squeezing operations employ pulses with many photons (n > 10), the linearization is always legitimate.
Based on the linearization approximation and the conservation of commutation brackets, a general, self-consistent scheme is developed to quantize nonlinear optical-pulse propagation problems. A general computation procedure is developed to calculate the quantum uncertainties of the inner product between any given function and the (perturbed) field operator. As an illustration, a self-consistent quantum theory of the self-Raman effect in optical fibers is presented. The influence of the self-Raman effect on soliton squeezing is examined.
We return to the notion of optical solitons as these were first introduced—namely as exact soliton and multi-soliton solutions of the semi-classical self-induced transparency (SIT) equations. The SIT equations, even in their most general inhomogeneously broadened forms, are infinite dimensional, completely integrable Hamiltonian systems, and thus display much mathematical structure, yet their exact soliton solutions display well the nonlinear (self-induced) transparency features of the physical phenomenon. The mathematical feature of complete integrability is handed down from the SIT equations to the nonlinear Schrödinger equations and from this point of view the nonlinear Schrödinger (NLS) equations equations are studied first of all at classical level and then at quantum level. The ‘quantum soliton’ of the quantum attractive NLS model is defined and its properties investigated in detail, particularly in a comparison with the particular properties ascribed to the ‘quantum soliton’ as described in a recently published article (Abraham 1999 Phys. World (February) 21) with this title; as suggested in that article, the quantum NLS soliton is seen as a qubit for quantum information purposes. The covariant form of the quantum attractive NLS model is the quantum sine-Gordon model and this is briefly studied in the context of quantum information also; new results for large area Josephson junctions are reported in this context. The quantum soliton of the one dimensional attractive NLS model has apparently been observed in 2002 in a Bose condensed metal vapour ( 7 Li, (Khaykovich et al 2002 Science 296 1290)) and a detailed calculation suggests that the condensate undergoes a ‘collapse’ from three to one space dimensions thus forming an exact soliton solution of the one dimensional quantum attractive NLS model.
Self-induced transparency in a resonant two-level system creates a 2\ensuremath{\pi}-soliton pulse, which produces large optical nonlinearities with a very fast response time and an extremely small loss. The self-phase modulation of a 2\ensuremath{\pi} soliton and the mutual-phase modulation by the collision of two 2\ensuremath{\pi} solitons in the excitonic range of the spectrum in CdS can achieve the effective ${\ensuremath{\chi}}^{(3)}$ coefficient of ${10}^{\mathrm{\ensuremath{-}}6}$ to ${10}^{\mathrm{\ensuremath{-}}4.5}$ esu for a pulse duration of 1 to 3 ps. The quantum theory of 2\ensuremath{\pi} solitons predicts this new nonlinear process can realize squeezed states and quantum nondemolition measurement 20 dB below the standard quantum limit.
Linearization of the self-induced transparency equations reduces the problem to a single linear integro-differential equation for the perturbation field of a soliton. The adjoint system is introduced to establish orthogonality relations that lead to the relations between the perturbed field and perturbed scattering data. The commutation relations of the perturbation field provide the quantization rules for the perturbed ``scattering data.'' The quantized solitons are described by operators for photon number, phase, momentum, and position. The evolution and detection of these fluctuations are studied. The contribution of the continuum part can be suppressed by suitable detection schemes.
A general quantum theory of self-induced transparency (SIT) solitons is developed with nonlinear quantum effects of atoms taken into account. The quantization starts from the coarse-grain-averaged light-atom interaction Hamiltonian by which the quantum effects of ensemble atoms are modeled. The calculation of quantum properties is performed by the backpropagation method, which takes into account the contribution of the field continuum parts and the atomic fluctuations. The detectable squeezing ratio with a general homodyne local oscillator can be simulated for practical experimental realizations. Schemes for generating quantum correlation through SIT soliton interaction are also proposed and analyzed.
A linearized quantum theory of soliton squeezing and detection is presented. The linearization reduces the quantum problem to a classical one. The classical formulation provides physical insight. It is shown that a quantized soliton exhibits uncertainties in photon number and phase, position (time), and momentum (frequency). Detectors for the measurement of all four operators are discussed. The squeezing of the soliton in the fiber is analyzed. An optimal homodyne detector for detection of the squeezing is presented that suppresses the noise associated with the continuum and the uncertainties in position and momentum.